Our theoretical analysis requires assumptions regarding the subsampling procedure, the event process, and the functional data. We state these assumptions and then our main theorems. We start by assuming there exists a \(\tau < \infty \)  such that all individuals are no longer at risk (i.e., \(\tau _i < \tau \)  for all \(i\) ). Moreover, define \(R_i (t)\)  to be the at-risk indicator for participant \(i\) , i.e., \(R_i (t) = {\bf 1} [t \in (0,\tau _i)]\) . Asymptotic theory provided in Lemma REF  will be proven under regularity conditions A-E in [1]} along with the following additional assumptions:
